class: center, middle, inverse, title-slide .title[ # Class 5a: Multiple Linear Regression ] .author[ ### Business Forecasting ] --- <style type="text/css"> .remark-slide-content { font-size: 20px; } </style> --- ## Roadmap ### This set of classes - What is a multiple linear regression --- ### Motivation - Suppose that you are administering a hospital - You need to know how many doctors, nurses and beds you need - So you want to predict how long a patient will stay at the urgent care -- - You collect the data on - The Duration of the visit - The type of patient - How many other people there are currently at urgent care - What kind of problem they came with - What type of bed they got -- - If we know these factors, can we predict how long patient will stay? --- ### Data <div class="datatables html-widget html-fill-item-overflow-hidden html-fill-item" id="htmlwidget-09657774999baae15364" style="width:100%;height:auto;"></div> <script type="application/json" 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ESPECIFICADO","MÉDICA","MÉDICA","MÉDICA"]],"container":"<table class=\"display\">\n <thead>\n <tr>\n <th>ID<\/th>\n <th>Duration<\/th>\n <th>Occupancy<\/th>\n <th>SEXO<\/th>\n <th>EDAD<\/th>\n <th>TIPOCAMA<\/th>\n <th>MOTATE<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"pageLength":10,"searching":false,"initComplete":"function(settings, json) {\n$(this.api().table().container()).css({'font-size': '12px'});\n}","columnDefs":[{"className":"dt-right","targets":[0,1,2,4]}],"order":[],"autoWidth":false,"orderClasses":false},"selection":{"mode":"multiple","selected":null,"target":"row","selectable":null}},"evals":["options.initComplete"],"jsHooks":[]}</script> --- ### Multiple linear regression Suppose that the outcome `\(y_i\)` (duration) is a linear function of `\(x_1\)` (occupancy) and `\(x_2\)` (age) `$$y_i=\beta_0+\beta_1x_{i1}+\beta_2x_{i2}+u_i$$` - `\(\beta_0\)` represents the value of `\(y_i\)` when `\(x_1\)` and `\(x_2\)` are 0. - `\(\beta_1\)` represents the change in `\(y_i\)` while changing `\(x_1\)` by one unit and keeping `\(x_2\)` constant - `\(\beta_2\)` represents the change in `\(y_i\)` while changing `\(x_2\)` by one unit and keeping `\(x_1\)` constant --- ### Multiple linear regression 100 observations simulated from an a regression line: `$$y_i=5+2x_{i1}+1x_{i2}+u_i$$` <div id="htmlwidget-1b580025d58e8b55ee51" style="width:80%;height:400px;" class="plotly html-widget "></div> <script type="application/json" data-for="htmlwidget-1b580025d58e8b55ee51">{"x":{"visdat":{"1c7c2e8946a8":["function () 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12.5366542090647"],"hoverinfo":"text","mode":"markers","type":"surface","opacity":0.59999999999999998,"frame":null}],"highlight":{"on":"plotly_click","persistent":false,"dynamic":false,"selectize":false,"opacityDim":0.20000000000000001,"selected":{"opacity":1},"debounce":0},"shinyEvents":["plotly_hover","plotly_click","plotly_selected","plotly_relayout","plotly_brushed","plotly_brushing","plotly_clickannotation","plotly_doubleclick","plotly_deselect","plotly_afterplot","plotly_sunburstclick"],"base_url":"https://plot.ly"},"evals":[],"jsHooks":[]}</script> --- ### Multiple linear regression 100 observations simulated from an a regression line: `$$y_i=5+2x_{i}-1x_i^2+u_i$$` <img src="data:image/png;base64,#C_5_slides_a_files/figure-html/unnamed-chunk-3-1.png" width="100%" /> --- ### Multiple linear regression Suppose that: `$$x_1 = \begin{cases} 1 & \text{if female} \\ 0 & \text{if male} \end{cases}$$` 100 observations simulated from an a regression line: `$$y_i=5+2x_{i1}-1x_{i2}+u_i$$` <img src="data:image/png;base64,#C_5_slides_a_files/figure-html/unnamed-chunk-4-1.png" width="100%" /> --- ### Multiple linear regression Now imagine a regression with k variables: `$$y_i=\beta_0+\beta_1x_{i1}+\beta_2x_{i2}+...+\beta_kx_{ik}+u_i$$` - Maybe you are trying to predict customer spending based on what they looked at and `\(x_{ij}\)` represent how long customer `\(i\)` looked at item `\(j\)` -- - Maybe you are trying to predict sales in a store `\(i\)`, and `\(x_{ij}\)` represent prices of the products, their competitors' products, how many people live around and how rich are they etc... -- - We can no longer visualize it (because we can't visualize more than 3 dimensions) --- ### Multiple linear regression We can also write it in the vector form: `$$y_i=\beta_0+\beta_1x_{i1}+\beta_2x_{i2}+...+\beta_k,x_{ik}+u_i$$` In vector form is: `$$\mathbf{y}=\mathbf{X\beta}+\mathbf{u}$$` <div class="math"> \[ \underbrace{\begin{bmatrix} y_1 \\ y_2 \\ \vdots \\ y_n \\ \end{bmatrix}}_{\substack{\mathbf{y} \\ n \times 1}} = \underbrace{\begin{bmatrix} 1 & x_{11} & x_{12} & ... & x_{1k} \\ 1 & x_{21} & x_{22} & ... & x_{2k} \\ \vdots & \vdots & \vdots & ....& \vdots \\ 1 & x_{n1} & x_{n2} & ... & x_{nk} & \end{bmatrix}}_{\substack{\mathbf{X} \\ n \times (k+1)}} \underbrace{\begin{bmatrix} \beta_0 \\ \beta_1 \\ \vdots \\ \beta_k \\ \end{bmatrix}}_{\substack{\mathbf{\beta} \\ (k+1) \times 1}} + \underbrace{\begin{bmatrix} u_1 \\ u_2 \\ \vdots \\ u_n \\ \end{bmatrix}}_{\substack{\mathbf{u} \\ n \times 1}} \] </div> --- ### Full Rank Important Assumption: **X is full rank** - Has same rank as the number of parameters: `\(p=k+1\)` - Also known as: no perfect multicolinearity -- - .blue[Technically]: columns of X should be linearly independent -- - .blue[Intuitively]: none of the variables are perfectly correlated. If they are perfectly correlated, then we don't need one of the columns because we can perfectly predict one column with information from another column. - Suppose that one column is income in USD, and the second one is income measured in Pesos. They are perfectly correlated. Once we know income in USD, income in Pesos does not bring any additional information. We would not be able to estimate the effect of both income in USD and income in Pesos at the same time. <div class="math"> \[ \begin{array}{cc} \text{Full Rank Matrix:} & \text{Matrix Not of Full Rank:} \\ \left[\begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{array}\right] & \left[\begin{array}{ccc} 1 & 2 & 4 \\ 4 & 5 & 10 \\ 7 & 8 & 16 \end{array}\right] \end{array} \] </div> --- ### Multiple linear regression **Goal:** - Estimate the vector of parameters `\(\mathbf{\beta}\)` **Procedure** - Find <div class="math"> \[ \mathbf{b}=\begin{bmatrix} b_0 \\ b_1 \\ \vdots \\ b_k \\ \end{bmatrix} \] </div> - Which minimizes the squared errors in the problem: `$$y_i=b_0+b_1x_{i1}+b_2x_{i2}+...+b_kx_{ik}+e_i$$` - That is minimize `$$SSE=\sum_ie_i^2=\sum_i(y_i-\hat{y}_i)^2=\mathbf{e}'\mathbf{e}=(\mathbf{y-\hat{y}})'(\mathbf{y-\hat{y}})=(\mathbf{y-X\hat{\beta}})'(\mathbf{y-X\hat{\beta}})$$` --- ### Multiple linear regression - We can do it with scalars `$$\begin{align*} \frac{\partial SSE}{\partial \hat{\beta}_0} & = -2\sum_{i=1}^{n} \left( y_i - (\hat{\beta}_0 + \hat{\beta}_1 x_{i1}+...+\hat{\beta}_k x_{ik})\right)=0 \\ \frac{\partial SSE}{\partial \hat{\beta}_1} & = -2\sum_{i=1}^{n} x_{i1} \left( y_i - (\hat{\beta}_0 + \hat{\beta}_1 x_{i1}+...+\hat{\beta}_k x_{ik})\right)=0 \\ \vdots \\ \frac{\partial SSE}{\partial \hat{\beta}_k} & = -2\sum_{i=1}^{n} x_{ik} \left( y_i - (\hat{\beta}_0 + \hat{\beta}_1 x_{i1}+...+\hat{\beta}_k x_{ik})\right)=0 \\ \end{align*}$$` -- - We have `\(k\)` equations with `\(k\)` unknowns. --- ### Multiple linear regression - Or we can do it with vectors -- - First rewrite the sum of squares: `$$SSE(b)=(\mathbf{y-Xb})'(\mathbf{y-Xb})=\mathbf{y'}\mathbf{y-2b'X'y}+\mathbf{b'X'Xb}$$` -- - Then minimize it with respect to `\(\mathbf{b}\)` `$$\frac{\partial}{\partial \mathbf{b}}\mathbf{y'}\mathbf{y-2b'X'y}+\mathbf{b'X'Xb}=\mathbf{-2X'y}+\mathbf{2X'Xb}$$` -- - `\(\hat{\beta}\)` is the solution of such minimization (our OLS estimator) `$$\begin{align*} \mathbf{-2X'y}+\mathbf{2X'X\hat{\beta}}&=0 \\ \mathbf{X'X\hat{\beta}} & =\mathbf{X'y} \\ \mathbf{\hat{\beta}} & =\mathbf{(X'X)^{-1}X'y} \end{align*}$$` --- ### Multiple linear regression Looking more closely at the **first order condition**: <div class="math"> \[ \underbrace{\begin{bmatrix} n & \sum_{i=1}^{n}x_{i1} & \ldots & \sum_{i=1}^{n} x_{ik} \\ \sum_{i=1}^{n} x_{i1} & \sum_{i=1}^{n} x_{i1}^2 & \ldots & \sum_{i=1}^{n} x_{i1}x_{ik} \\ \vdots & \vdots & \ddots & \vdots \\ \sum_{i=1}^{n} x_{ik}& \sum_{i=1}^{n} x_{ik}x_{i1} & \ldots & \sum_{i=1}^{n} x_{ik}^2\end{bmatrix}}_{\mathbf{X'X}} \underbrace{\begin{bmatrix} \hat{\beta}_0 \\ \hat{\beta}_1 \\ \vdots \\ \hat{\beta}_k\end{bmatrix}}_{\hat{\beta}} = \underbrace{\begin{bmatrix} \sum_{i=1}^{n}y_i \\ \sum_{i=1}^{n}x_{i1}y_i \\ \vdots \\ \sum_{i=1}^{n}x_{ik}y_i\end{bmatrix}}_{\mathbf{X'y}} \] </div> Looking more closely and it's **solution**: <div class="math"> \[\underbrace{\begin{bmatrix} \hat{\beta}_0 \\ \hat{\beta}_1 \\ \vdots \\ \hat{\beta}_k\end{bmatrix}}_{\hat{\beta}} = \underbrace{\begin{bmatrix} n & \sum_{i=1}^{n}x_{i1} & \ldots & \sum_{i=1}^{n} x_{ik} \\ \sum_{i=1}^{n} x_{i1} & \sum_{i=1}^{n} x_{i1}^2 & \ldots & \sum_{i=1}^{n} x_{i1}x_{ik} \\ \vdots & \vdots & \ddots & \vdots \\ \sum_{i=1}^{n} x_{ik}& \sum_{i=1}^{n} x_{ik}x_{i1} & \ldots & \sum_{i=1}^{n} x_{ik}^2\end{bmatrix}^{-1}}_{\mathbf{(X'X)}^{-1}} \underbrace{\begin{bmatrix} \sum_{i=1}^{n}y_i \\ \sum_{i=1}^{n}x_{i1}y_i \\ \vdots \\ \sum_{i=1}^{n}x_{ik}y_i\end{bmatrix}}_{\mathbf{X'y}} \] </div> --- ### Special Case: k=1 What if we have just one `\(x\)`? <div class="math"> \[\underbrace{\begin{bmatrix} \hat{\beta}_0 \\ \hat{\beta}_1 \end{bmatrix}}_{\hat{\beta}} = \underbrace{\begin{bmatrix} n & \sum_{i=1}^{n}x_{i1} \\ \sum_{i=1}^{n} x_{i1} & \sum_{i=1}^{n} x_{i1}^2\end{bmatrix}^{-1}}_{\mathbf{(X'X)}^{-1}} \underbrace{\begin{bmatrix} \sum_{i=1}^{n}y_i \\ \sum_{i=1}^{n}x_{i1}y_i \end{bmatrix}}_{\mathbf{X'y}} \] </div> -- <div class="math"> \[ \begin{bmatrix} \hat{\beta}_0 \\ \hat{\beta}_1\end{bmatrix} = \begin{bmatrix} \frac{\sum^n_{i=1}x_{i1}^2}{n\sum_{i=1}^{n} x_{i1}^2 - (\sum_{i=1}^{n} x_{i1})^2} & \frac{-\sum_{i=1}^{n} x_{i1}}{n\sum_{i=1}^{n} x_{i1}^2 - (\sum_{i=1}^{n} x_{i1})^2} \\ \frac{-\sum_{i=1}^{n} x_{i1}}{n\sum_{i=1}^{n} x_{i1}^2 - (\sum_{i=1}^{n} x_{i1})^2} & \frac{n}{n\sum_{i=1}^{n} x_{i1}^2 - (\sum_{i=1}^{n} x_{i1})^2}\end{bmatrix} \begin{bmatrix} \sum_{i=1}^{n}y_i \\ \sum_{i=1}^{n}x_{i1}y_i \end{bmatrix} \] </div> -- which gives: <div class="math"> \[ \begin{bmatrix} \hat{\beta}_0 \\ \hat{\beta}_1\end{bmatrix} = \begin{bmatrix} \bar{y}-\bar{x}_1\frac{\sum(x^2_{1i}-n\bar{y}\bar{x}_1)}{\sum_{i=1}^{n} x_{i1}^2 - n\bar{x}_{1}^2} \\ \frac{\sum_ix_iy_i-n\bar{x}_{1}\bar{y}}{\sum_{i=1}^{n} x_{i1}^2 - n\bar{x}_{1}^2}\end{bmatrix} \] </div> --- ### Predictions To make predictions based on the estimated regressors we use: `$$\hat{y}_i=\hat{\beta}_0+\hat{\beta}_1x_{i1}+\hat{\beta}_2x_{i2}+...+\hat{\beta}_kx_{ik}$$` Or in the vector form: `$$\mathbf{\hat{y}}=\mathbf{X\hat{\beta}}=\mathbf{X\mathbf{(X'X)}^{-1}\mathbf{X'y}}=\mathbf{Hy}$$` Where `\(\mathbf{H}=\mathbf{X(X'X)}^{-1}\mathbf{X}\)` is called a hat matrix. --- ### Residuals To get residuals, we calculate: `$$e_i=y_i-\hat{y}_i=y_i-\hat{\beta}_0+\hat{\beta}_1x_{i1}+\hat{\beta}_2x_{i2}+...+\hat{\beta}_kx_{ik}$$` Or in the vector form: `$$\mathbf{e}=\mathbf{y-\hat{y}}=y-\mathbf{X\hat{\beta}}=\mathbf{y}-\mathbf{X\mathbf{(X'X)}^{-1}\mathbf{X'y}}=\mathbf{(I-H)y}$$` --- #### Example with numbers `$$\small \begin{align*} \text{Dataset:} \\ &\begin{array}{|c|c|c|c|} \hline \text{Student} & \text{Hours Studied (}x_1\text{)} & \text{Hours Slept (}x_2\text{)} & \text{Exam Score (}y\text{)} \\ \hline 1 & 3 & 8 & 80 \\ 2 & 4 & 7 & 85 \\ 3 & 6 & 6 & 92 \\ 4 & 5 & 7 & 88 \\ \hline \end{array} \\ \text{X matrix}: \\ & X = \begin{bmatrix} 1 & 3 & 8 \\ 1 & 4 & 7 \\ 1 & 6 & 6 \\ 1 & 5 & 7 \\ \end{bmatrix} \\ \text{Response Vector } (y): \\ & y = \begin{bmatrix} 80 \\ 85 \\ 92 \\ 88 \\ \end{bmatrix} \end{align*}$$` We are trying to find: `$$\mathbf{\hat{\beta}} =\mathbf{(X'X)^{-1}X'y}$$` --- #### Example with numbers Multiply `\(X'\)` by `\(X\)`: `$$X'X = \begin{bmatrix} 1 & 1 & 1 & 1 \\ 3 & 4 & 6 & 5 \\ 8 & 7 & 6 & 7 \\ \end{bmatrix} \begin{bmatrix} 1 & 3 & 8 \\ 1 & 4 & 7 \\ 1 & 6 & 6 \\ 1 & 5 & 7 \\ \end{bmatrix}= \begin{bmatrix} 4 & 18 & 28 \\ 18 & 86 & 123 \\ 28 & 123 & 198 \\ \end{bmatrix}$$` Find the inverse `\((X'X)^{-1}\)` `$$(X'X)^{-1} = \begin{bmatrix} 474.75 & -30 & -48.5 \\ -30 & 2 & 3 \\ -48.5 & 3 & 5 \\ \end{bmatrix}$$` --- #### Example with numbers Next let's find `\(X'y\)` `$$X'y= \begin{bmatrix} 1 & 1 & 1 & 1 \\ 3 & 4 & 6 & 5 \\ 8 & 7 & 6 & 7 \\ \end{bmatrix} \begin{bmatrix} 80 \\ 85 \\ 92 \\ 88 \\ \end{bmatrix}= \begin{bmatrix} 345 \\ 1572 \\ 2403 \\ \end{bmatrix}$$` So, our coefficients are: `$$\beta=\begin{bmatrix} \beta_0 \\ \beta_1 \\ \beta_2 \\ \end{bmatrix} = \underbrace{\begin{bmatrix} 474.75 & -30 & -48.5 \\ -30 & 2 & 3 \\ -48.5 & 3 & 5 \\ \end{bmatrix}}_{(X'X)^{-1}} \underbrace{\begin{bmatrix} 345 \\ 1572 \\ 2403 \\ \end{bmatrix}}_{X'y}= \begin{bmatrix} 83.25 \\ 3 \\ -1.5 \\ \end{bmatrix}$$` -- **Interpretation** - Score with 0 hours of sleep and 0 of studying is 83.25 - 1 more hour of studying (without changing sleep hours) increases score by 3 - 1 more hour of sleep (without changing study hours) decreases score by 1.5 --- #### Example with numbers We can find predicted values: `$$\hat{y}=X\hat{\beta}= \begin{bmatrix} 1 & 3 & 8 \\ 1 & 4 & 7 \\ 1 & 6 & 6 \\ 1 & 5 & 7 \\ \end{bmatrix}\begin{bmatrix} 83.25 \\ 3 \\ -1.5 \\ \end{bmatrix}= \begin{bmatrix} 80.25 \\ 84.75 \\ 92.25 \\ 87.75 \\ \end{bmatrix}$$` And the residuals: `$$e=y-\hat{y}=y-X\hat{\beta}= \begin{bmatrix} 80 \\ 85 \\ 92 \\ 88 \\ \end{bmatrix}- \begin{bmatrix} 80.25 \\ 84.75 \\ 92.25 \\ 87.75 \end{bmatrix}= \begin{bmatrix} -0.25 \\ 0.25 \\ -0.25 \\ 0.25 \end{bmatrix}$$` --- #### Example from data: ```r # Fit a linear regression model lm_model <- lm(Duration ~ Occupancy+EDAD, data = Sample_urg) # Display the summary of the linear regression model summary(lm_model) ``` ``` ## ## Call: ## lm(formula = Duration ~ Occupancy + EDAD, data = Sample_urg) ## ## Residuals: ## Min 1Q Median 3Q Max ## -773.63 -26.60 -17.27 -0.54 1252.76 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 23.23646 2.48365 9.356 < 2e-16 *** ## Occupancy 3.70348 0.10088 36.711 < 2e-16 *** ## EDAD 0.20603 0.06743 3.055 0.00226 ** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 98.97 on 4997 degrees of freedom ## Multiple R-squared: 0.2169, Adjusted R-squared: 0.2166 ## F-statistic: 692 on 2 and 4997 DF, p-value: < 2.2e-16 ``` --- ### Correlations vs Coefficients Note, that `\(x_1\)` and `\(x_2\)` can both have positive correlation with `\(y_i\)`, but different coefficients! - Suppose `\(x_1\)` is study hours, `\(x_2\)` is coffee cups drunk by a student, and `\(y\)` is student's score on the exam. <img src="data:image/png;base64,#C_5_slides_a_files/figure-html/unnamed-chunk-6-1.png" width="1000%" /> --- ### Correlations vs Coefficients ``` ## ## Call: ## lm(formula = y ~ x1 + x2, data = data) ## ## Residuals: ## Min 1Q Median 3Q Max ## -2.779 -1.422 -0.418 1.096 6.305 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 3.13966 0.38033 8.255 7.68e-13 *** ## x1 2.06132 0.11686 17.640 < 2e-16 *** ## x2 -0.08510 0.09798 -0.868 0.387 ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 1.88 on 97 degrees of freedom ## Multiple R-squared: 0.9018, Adjusted R-squared: 0.8997 ## F-statistic: 445.2 on 2 and 97 DF, p-value: < 2.2e-16 ``` -- - Why coffee has 0 impact? -- - Because it only helps to study longer, but comparing students who study the same amount, drinking more coffee is not better. --- ### OLS Properties - As usual, we asked whether it's unbiased and what is its variance. -- - **Unbiased**: <div class="math"> \[\small \begin{align*} E(\hat{\beta}) & =E(\mathbf{(X'X)^{-1}X'y})=E(\mathbf{(X'X)^{-1}X'(X\beta+u)})) \\ & = E(\mathbf{(X'X)^{-1}X'(X\beta+u)}))=E(\mathbf{(X'X)^{-1}X'X\beta})+E(\mathbf{(X'X)^{-1}X'u}) \\ & = \beta+0=\beta \end{align*} \] </div> Where `\(\small E(\mathbf{(X'X)^{-1}X'u})\)` if `\(\small E(u|X)=0\)` (our usual assumption). -- - **Variance** `$$\small Var(\beta)=Cov(\beta)=\underbrace{\begin{bmatrix} var(\beta_0) & cov(\beta_0, \beta_1) & ... & cov(\beta_0, \beta_k) \\ cov(\beta_1, \beta_0) & var(\beta_1) & ... & cov(\beta_1, \beta_k) \\ \vdots & \vdots & \vdots & \vdots \\ cov(\beta_k, \beta_0) & cov(\beta_k, \beta_1) & ... & var(\beta_k) & \end{bmatrix}}_{\substack{ \\ (k+1) \times (k+1)}}$$` - So it's a matrix with variance of single parameters on the diagonal and covariances off the diagonal. --- ### Variance First, note that: `$$\hat{\beta} = (\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}'\mathbf{X}\beta + \mathbf{u} = \beta + (\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}'\mathbf{u}$$` Let's use this <div class="math"> \[\small \begin{align*} var(\hat{\beta}) & = \mathbb{E}[(\hat{\beta} - \mathbb{E}[\hat{\beta}]) (\hat{\beta} - \mathbb{E}[\hat{\beta}])'] \\ & = \mathbb{E}[(X'X)^{-1}X'\mathbf{u} ((X'X)^{-1}X'\mathbf{u})']= (X'X)^{-1}X'\mathbb{E}[\mathbf{u}\mathbf{u}']X(X'X)^{-1} \\ & = (X'X)^{-1}X'(I\sigma^2)X(X'X)^{-1} =\sigma^2(X'X)^{-1} \end{align*} \] </div> -- So `$$var(\hat{\beta}_k)=\sigma^2(X'X)^{-1}_{kk}$$` where `\((X'X)^{-1}_{kk}\)` is element in `\(k\)` row and `\(k\)` column of `\((X'X)^{-1}\)` matrix. -- - And standard deviation is just square root of this! -- **Intuition** If we have just one regressor: `\((X'X)^{-1}_{11}=\frac{1}{\sum(x_i-\bar{x})^2}\)` --- ### Variance - Where the hell do we get the `\(\sigma^2\)` from?! -- - Same as before: `$$\sigma^2=\frac{\sum_i e_i^2}{n-p}$$` - Where `\(e_i\)` is fitted residual and `\(p\)` is number of parameters `\(p=k+1\)` - This is called mean squared error as well -- The easiest way to compute this sum is: `$$\sum_i e_i^2=\mathbf{e'}\mathbf{e}=(\mathbf{y-X\hat{\beta}})'(\mathbf{y-X\hat{\beta}})=\mathbf{y'y-\hat{\beta}'X'y}$$` --- #### Gauss Markov Theorem (Again) Assumptions - `\(E(u_i|X)=0\)` - `\(var(u_i)=\sigma^2\)` - `\(cov(u_i,u_j)=0\)` - `\(X\)` is full rank NO NEED FOR NORMALITY **Theorem:** OLS is .blue[BLUE:] Best, Linear, Unbiased Estimator - It has the lowest variance among linear and unbiased estimators -- - What's a linear estimator? - It's an estimator where `\(\beta\)` coefficients are linear functions of outcomes - Anything of the form `\(b=Cy\)` where C is p x n matrix. - So `\(b_1=c_{11}y_1+c_{12}y_2+...+c_{13}y_3\)` - Example `\(b_1=\frac{1}{n}y_1+...+\frac{1}{n}y_n\)` -- - How is OLS linear? `\(\hat{\beta}=Cy=\underbrace{(X'X)^{-1}X'}_{C}y\)` --- ### Categorical Variables in a Regression - Suppose we want to learn whether mode of work affects workers productivity. - Each worker can be in one of these 3 categories: - Fully at the office - Fully remote - Hybrid <div class="datatables html-widget html-fill-item-overflow-hidden html-fill-item" id="htmlwidget-8d1c30200779549e2ea8" style="width:100%;height:auto;"></div> <script type="application/json" data-for="htmlwidget-8d1c30200779549e2ea8">{"x":{"filter":"none","vertical":false,"fillContainer":false,"data":[[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100],[93,75,108,115,75,88,107,115,125,99,104,114,121,125,73,100,91,121,118,77,113,84,74,126,79,109,80,71,101,75,87,95,105,118,82,98,97,114,113,120,123,127,103,101,80,99,85,72,108,102,78,87,105,80,76,96,91,121,86,71,122,90,108,103,93,81,96,108,101,110,114,99,93,70,70,130,76,95,113,115,122,106,115,121,107,118,71,95,122,109,125,80,89,118,129,89,70,130,79,73],["Fully remote","Fully remote","Hybrid","Hybrid","Hybrid","Hybrid","Fully remote","Fully at the office","Fully at the office","Hybrid","Fully at the office","Fully remote","Fully remote","Hybrid","Fully remote","Fully at the office","Hybrid","Fully at the office","Fully at the office","Hybrid","Fully remote","Fully at the office","Fully at the office","Fully remote","Fully remote","Fully remote","Fully remote","Fully at the office","Hybrid","Hybrid","Fully remote","Hybrid","Hybrid","Hybrid","Fully remote","Hybrid","Hybrid","Fully remote","Fully remote","Hybrid","Fully remote","Hybrid","Fully remote","Fully remote","Fully at the office","Fully remote","Hybrid","Fully at the office","Fully at the office","Hybrid","Fully remote","Fully at the office","Fully at the office","Hybrid","Hybrid","Hybrid","Fully at the office","Fully remote","Fully remote","Hybrid","Hybrid","Hybrid","Hybrid","Fully remote","Fully at the office","Hybrid","Fully remote","Fully at the office","Hybrid","Hybrid","Hybrid","Fully remote","Fully at the office","Fully remote","Hybrid","Fully at the office","Fully remote","Fully at the office","Fully remote","Fully remote","Hybrid","Fully remote","Fully remote","Fully remote","Fully remote","Fully remote","Hybrid","Fully at the office","Fully at the office","Hybrid","Hybrid","Hybrid","Hybrid","Fully remote","Fully at the office","Fully at the office","Hybrid","Fully remote","Fully remote","Hybrid"]],"container":"<table class=\"display\">\n <thead>\n <tr>\n <th>WorkerID<\/th>\n <th>Productivity<\/th>\n <th>WorkMode<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"pageLength":8,"searching":false,"initComplete":"function(settings, json) {\n$(this.api().table().container()).css({'font-size': '12px'});\n}","columnDefs":[{"className":"dt-right","targets":[0,1]}],"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[8,10,25,50,100]},"selection":{"mode":"multiple","selected":null,"target":"row","selectable":null}},"evals":["options.initComplete"],"jsHooks":[]}</script> --- - How do we estimate the impact of categorical variable? - We turn it into a series of binary variables (or indicator variables)! `$$D_{i, Remote}=\begin{cases} 1 & WorkMode_i=Fully Remote \\ 0 & otherwise \end{cases}$$` `$$D_{i,Hybrid}=\begin{cases} 1 & WorkMode_i=Hybrid\\ 0 & otherwise \end{cases}$$` <div class="datatables html-widget html-fill-item-overflow-hidden html-fill-item" id="htmlwidget-a0b5fef8fc183121e9a3" style="width:100%;height:auto;"></div> <script type="application/json" data-for="htmlwidget-a0b5fef8fc183121e9a3">{"x":{"filter":"none","vertical":false,"fillContainer":false,"data":[[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100],[112,124,108,76,125,111,79,115,127,95,88,86,83,104,115,110,103,119,116,71,93,73,118,125,103,121,105,110,101,116,124,119,74,70,73,122,105,89,128,105,102,93,89,119,73,92,121,112,111,91,103,78,117,123,82,116,106,127,80,102,122,122,71,129,99,93,95,76,80,94,91,95,88,104,79,127,73,92,128,109,74,95,96,100,97,107,78,124,104,103,77,116,99,117,73,119,86,87,109,127],["Fully at the office","Hybrid","Hybrid","Fully at the office","Fully remote","Fully at the office","Fully remote","Fully remote","Fully at the office","Hybrid","Fully at the office","Fully remote","Fully remote","Fully at the office","Fully remote","Fully at the office","Fully at the office","Fully remote","Fully remote","Fully remote","Fully remote","Hybrid","Fully at the office","Fully at the office","Fully at the office","Fully at the office","Hybrid","Fully remote","Fully at the office","Fully at the office","Hybrid","Hybrid","Fully at the office","Fully at the office","Fully remote","Hybrid","Fully at the office","Fully at the office","Hybrid","Fully at the office","Fully remote","Fully remote","Hybrid","Fully at the office","Hybrid","Fully at the office","Hybrid","Hybrid","Fully at the office","Hybrid","Hybrid","Hybrid","Hybrid","Hybrid","Fully remote","Hybrid","Fully remote","Hybrid","Fully remote","Fully at the office","Fully at the office","Hybrid","Fully at the office","Fully remote","Fully at the office","Fully remote","Fully remote","Fully at the office","Fully at the office","Fully remote","Hybrid","Fully at the office","Fully remote","Fully at the office","Fully remote","Hybrid","Fully remote","Fully remote","Hybrid","Fully at the office","Fully remote","Hybrid","Hybrid","Fully at the office","Hybrid","Fully at the office","Fully at the office","Hybrid","Fully at the office","Fully at the office","Hybrid","Fully at the office","Fully at the office","Fully at the office","Fully remote","Fully remote","Fully remote","Fully remote","Hybrid","Hybrid"],[1,0,0,1,0,1,0,0,1,0,1,0,0,1,0,1,1,0,0,0,0,0,1,1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,1,0,0,0,1,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,1,1,0,1,0,1,0,0,1,1,0,0,1,0,1,0,0,0,0,0,1,0,0,0,1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0],[0,0,0,0,1,0,1,1,0,0,0,1,1,0,1,0,0,1,1,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,1,0,0,0,0,1,0,1,1,0,0,1,0,0,1,0,1,0,1,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,0,0],[0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,1,1,0,0,0,1,0,0,1,0,0,0,1,0,1,0,1,1,0,1,1,1,1,1,0,1,0,1,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,1,0,0,1,1,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,1,1]],"container":"<table class=\"display\">\n <thead>\n <tr>\n <th>WorkerID<\/th>\n <th>Productivity<\/th>\n <th>WorkMode<\/th>\n <th>WorkModeFully.at.the.office<\/th>\n <th>WorkModeFully.remote<\/th>\n <th>WorkModeHybrid<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"pageLength":6,"searching":false,"initComplete":"function(settings, json) {\n$(this.api().table().container()).css({'font-size': '12px'});\n}","columnDefs":[{"className":"dt-right","targets":[0,1,3,4,5]}],"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[6,10,25,50,100]},"selection":{"mode":"multiple","selected":null,"target":"row","selectable":null}},"evals":["options.initComplete"],"jsHooks":[]}</script> -- - For each person, only one of these dummies is equal to 1! --- - We will add these dummies into a regression, but not all of them! - If we have m categories, we will add m-1 dummies. Why? `$$y_i=\beta_0+\beta_1D_{i1}+\beta_2D_{i2}+...+\beta_{m-1}D_{im-1}+u_i$$` - In our Example: `$$y_i=\beta_0+\beta_1D_{i,Hybrid}+\beta_2D_{i,Remote}+u_i$$` -- - Because otherwise X would not be full rank! <div class="math"> \[ \begin{array}{cc} \text{Full Rank Matrix:} & \text{Matrix Not of Full Rank:} \\ \left[\begin{array}{ccc} 1 & 1 & 0 \\ 1 & 0 & 0 \\ 1 & 0 & 1 \end{array}\right] & \left[\begin{array}{cccc} 1 & 1 & 0 & 0\\ 1 & 0 & 0 & 1\\ 1 & 0 & 1 & 0 \end{array}\right] \end{array} \] </div> -- - Intuitively, if I know that the values of `\(D_{i,Hybrid}\)` and `\(D_{i,Remote}\)`, I know the value of `\(D_{i,Office}\)` - Ex: if they don't work hybrid and don't work remote, I know they work at the office - So including it does not bring any new information --- - R automatically transform categorical variable to dummies and excludes one of them ```r # Fit a linear regression model lm_model <- lm(Productivity ~ WorkMode, data = d) # Display the summary of the linear regression model summary(lm_model) ``` ``` ## ## Call: ## lm(formula = Productivity ~ WorkMode, data = d) ## ## Residuals: ## Min 1Q Median 3Q Max ## -34.774 -12.636 0.946 14.410 34.667 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 101.590 2.695 37.697 <2e-16 *** ## WorkModeFully remote -7.256 4.087 -1.775 0.079 . ## WorkModeHybrid 6.184 4.050 1.527 0.130 ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 16.83 on 97 degrees of freedom ## Multiple R-squared: 0.09125, Adjusted R-squared: 0.07251 ## F-statistic: 4.87 on 2 and 97 DF, p-value: 0.009652 ``` --- ### Interpretation of Coefficients - Coefficient on a dummy `\(D_1\)` tells us by how much `\(y\)` changes when we change category from the excluded one to the category 1. - In our example - Excluded category is: work fully at the office - this is our comparision group - `\(\beta_{hybrid}=6.184\)`: employees working in hybrid mode have on average 6.184 higher productivity score compared to the ones working at the office - `\(\beta_{remote}=-7.256\)`: employees working in fully remotely have on average 7.256 lower productivity score compared to the ones working at the office - The t-test on these coefficients tells us whether these differences in means across categories are significant! - Bottom line: the coefficients on the dummies show the average difference between `\(y\)` in that category compared to the excluded category (holding everything else unchanged) --- ### Example Suppose we have a categorical variable representing education level. 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Error t value Pr(>|t|) ## (Intercept) 70342 3125 22.509 < 2e-16 *** ## EducationPhD 14639 4008 3.652 0.000424 *** ## EducationMaster 22303 4157 5.365 5.59e-07 *** ## EducationBachelor 16993 4273 3.977 0.000135 *** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 13980 on 96 degrees of freedom ## Multiple R-squared: 0.2401, Adjusted R-squared: 0.2164 ## F-statistic: 10.11 on 3 and 96 DF, p-value: 7.517e-06 ``` --- ### Interactions Consider a regression: $$\text{Duration}_i=\beta_0+\beta_1\text{Occupancy}_i+\beta_2\text{Male}_i+u_i $$ - Where Male is a for patient `\(i\)` being male - We assumed that occupancy has always the same effect, independent of your gender -- - But what if occupancy matters more for men? - In other words: one additional patient on urgent care increases duration by more if you are a men? - Why? Maybe because when there is a lot of patients, doctors prioritize women (or men) -- - We want allow the coefficient on occupancy to differ by gender. How? --- ### Interactions - Run the regression: `$$\text{Duration}_i=\beta_0+\beta_1\text{Occupancy}_i+\beta_2\text{Male}_i+\beta_3\text{Occupancy}_i*\text{Male}_i +u_i$$` -- - What's the coefficient on Occupancy when you are a woman `\(Male_i=0\)`? `$$\begin{align*} & \text{Duration}_i=\beta_0+\beta_1\text{Occupancy}_i+\beta_2\text{Male}_i+\beta_3\text{Occupancy}_i*0 +u_i \\ & \text{Duration}_i=\beta_0+\beta_1\text{Occupancy}_i+\beta_2\text{Male}_i+u_i \end{align*}$$` -- - What's the coefficient on Occupancy when you are a man `\(Male_i=1\)`? `$$\begin{align*} & \text{Duration}_i=\beta_0+\beta_1\text{Occupancy}_i+\beta_2\text{Male}_i+\beta_3\text{Occupancy}_i*1 +u_i \\ & \text{Duration}_i=\beta_0+(\beta_1+\beta_3)\text{Occupancy}_i+\beta_2\text{Male}_i+u_i \end{align*}$$` -- We can estimate `\(\beta_3\)` and it will tell us by how much bigger is the coefficient on occupancy for men compared to the coefficient on occupancy for women. - `\(\beta_1\)` is the coefficient for women - `\(\beta_1+\beta_3\)` is the coefficient for women - `\(\beta_3\)` is the difference in slopes, which we can test like other coefficients --- ``` ## ## Call: ## lm(formula = Duration ~ Occupancy * SEXO, data = Sample_urg[Sample_urg$SEXO != ## "NO ESPECIFICADO", ]) ## ## Residuals: ## Min 1Q Median 3Q Max ## -1030.01 -26.49 -17.87 -1.11 1297.28 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 30.8015 1.8861 16.331 <2e-16 *** ## Occupancy 2.6903 0.1264 21.278 <2e-16 *** ## SEXOMASCULINO -4.8637 3.2324 -1.505 0.132 ## Occupancy:SEXOMASCULINO 2.6174 0.2031 12.889 <2e-16 *** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 97.27 on 4994 degrees of freedom ## Multiple R-squared: 0.2441, Adjusted R-squared: 0.2437 ## F-statistic: 537.6 on 3 and 4994 DF, p-value: < 2.2e-16 ``` - One additional patients increases duration for women by 2.69 minutes - One additional patients increases duration for men by 2.69+2.61=5.2 minutes -- - What could be reasons for this? --- ### Interactions - More generally, we can rewrite a regression: `$$y_i=\beta_0+\beta_1x_{i1}+\beta_2x_{i2}+\beta_3x_{i1}*x_{i2}+u_i$$` - As `$$y_i=\beta_0+(\beta_1+\beta_3x_{i2})x_{i1}+\beta_2x_{i2}++u_i$$` -- - `\(\beta_3\)` answers the following question: - If I increase `\(x_{i2}\)` by one, by how much the coefficient on `\(x_i1\)` changes? --- ### Interactions - Suppose you want to know who benefits the most from working from home. You collect survey data for each employee on the job satisfaction, whether they work in the office or from home, and the distance between the office and home -- - Who do you think benefits most from working from home? -- - How would you test this? -- `$$\text{Satisfaction}_i=\beta_0+\beta_1\text{WFH}_i+\beta_2\text{Distance}_i+\beta_3\text{WFH}_i*\text{Distance}_i +u_i$$` -- - What's the interpretation of `\(\beta_3\)`? -- - By how much the effect of working from home on satisfaction changes when we increase distance by one unit (km) -- - Which sign do you expect `\(\beta_3\)` to have? --- ### Goodness of fit - We can use again the R square to measure the goodness of fit. `$$\small R^2=1-\frac{\sum(y_i-\hat{y}_i)^2}{\sum(y_i-\bar{y}_i)^2}$$` -- - However, there is one problem with it. - Even if we add variables unrelated to `\(y\)`, the `\(R^2\)` would typically still increase by a bit -- - Even if in population there is 0 relationship with this variable, our sample is small so we will never get exactly 0 relationship -- - Sampling noise will make coefficient slightly positive or negative -- - So the increase in `\(R^2\)` will reflect that noise in our sample -- - The more coefficients we include, the higher `\(R^2\)` - We can adjust it, by accounting for the number of parameters used -- `$$\small R_{Adj}^2=1-\frac{\sum(y_i-\hat{y}_i)^2/(n-p)}{\sum(y_i-\bar{y}_i)^2/(n-1)}$$` -- - More parameters -> `\(\downarrow(n-p)\rightarrow\uparrow\sum(y_i-\hat{y}_i)^2/(n-p)\rightarrow\downarrow R_{Adj}^2\)` - So it balances off the mechanical effect of higher `\(R^2\)` due to more regressors --- ``` ## ## Call: ## lm(formula = Duration ~ Occupancy + EDAD, data = Sample_urg[Sample_urg$SEXO != ## "NO ESPECIFICADO", ]) ## ## Residuals: ## Min 1Q Median 3Q Max ## -773.65 -26.61 -17.27 -0.57 1252.75 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 23.23422 2.48416 9.353 < 2e-16 *** ## Occupancy 3.70354 0.10090 36.705 < 2e-16 *** ## EDAD 0.20626 0.06747 3.057 0.00225 ** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 98.99 on 4995 degrees of freedom ## Multiple R-squared: 0.2169, Adjusted R-squared: 0.2166 ## F-statistic: 691.8 on 2 and 4995 DF, p-value: < 2.2e-16 ``` --- ``` ## ## Call: ## lm(formula = Duration ~ Occupancy + EDAD + Random_var, data = Sample_urg[Sample_urg$SEXO != ## "NO ESPECIFICADO", ]) ## ## Residuals: ## Min 1Q Median 3Q Max ## -772.35 -26.71 -17.26 -0.64 1251.42 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 26.03637 3.90155 6.673 2.77e-11 *** ## Occupancy 3.70448 0.10091 36.712 < 2e-16 *** ## EDAD 0.20469 0.06749 3.033 0.00243 ** ## Random_var -0.50250 0.53950 -0.931 0.35168 ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 98.99 on 4994 degrees of freedom ## Multiple R-squared: 0.217, Adjusted R-squared: 0.2166 ## F-statistic: 461.4 on 3 and 4994 DF, p-value: < 2.2e-16 ``` - Adding random variable increased `\(R^2\)` but not `\(R^2_{Adj}\)` --- layout: false class: inverse, middle # Statistical Properties of OLS --- ### Inference - Let's add the assumption that errors are normally distributed: $$ \mathbf{u} \sim N(0,\sigma I) $$ Which means that: $$y \sim N(X\beta,\sigma I) $$ - With inference we can: - Do hypothesis testing on single coefficients, ex: `\(H_0: \beta_2=0\)` - Find confidence intervals for a single coefficients - Do hypothesis testing on multiple coefficients: ex: `\(H_0: \beta_1=\beta_2\)` --- ### Test for a Single Coefficient Under the above assumptions: `$$\hat{\beta}\sim N(\beta, \sigma\sqrt{(X'X)^{-1}})$$` And `$$\hat{\beta_j}\sim N(\beta, \sigma\sqrt{(X'X)^{-1}_{jj}})$$` -- Normalizing we get that: `$$\frac{\hat{\beta}_j-\beta_j}{s\sqrt{(X'X)^{-1}_{jj}}} \sim t_{n-p}$$` - This test statistic has student t distribution with n-p degrees of freedom - Because the `\(\frac{s^2(n-p)}{\sigma^2} \sim \chi_{n-p}\)` - Where p is the number of parameters (coefficients) - `\(p=k+1\)`: k regressors and 1 intercept --- ### Test for a single coefficient Suppose: - `\(H_0: \beta_j=\beta_{j0}\)` - `\(H_A: \beta_j \neq \beta_{j0}\)` Then, we use test statistic: `$$t_{test}=\frac{\hat{\beta}_j-\beta_{j0}}{s\sqrt{(X'X)^{-1}_{jj}}}$$` And we reject if `\(t_{test}>t_{\alpha/2,n-p}\)` or `\(t_{test}<-t_{\alpha/2,n-p}\)` Where `\(t_{\alpha/2,n-p}\)` is `\(1-\alpha/2\)` quantile of student t with n-p degrees of freedom --- ### Example Suppose: - `\(H_0: \beta_{Age}=0\)` - `\(H_A: \beta_{Age} \neq 0\)` ``` ## ## Call: ## lm(formula = Duration ~ Occupancy + EDAD, data = Sample_urg) ## ## Residuals: ## Min 1Q Median 3Q Max ## -773.63 -26.60 -17.27 -0.54 1252.76 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 23.23646 2.48365 9.356 < 2e-16 *** ## Occupancy 3.70348 0.10088 36.711 < 2e-16 *** ## EDAD 0.20603 0.06743 3.055 0.00226 ** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 98.97 on 4997 degrees of freedom ## Multiple R-squared: 0.2169, Adjusted R-squared: 0.2166 ## F-statistic: 692 on 2 and 4997 DF, p-value: < 2.2e-16 ``` --- ### Confidence Interval for a Single Coefficient We can also use this distribution to construct confidence intervals: An interval for `\(\beta_j\)` with confidence level `\(1-\alpha\)` is: `$$\begin{align*} CI_{1-\alpha} & =\{\hat{\beta_j}-t_{\alpha/2,n-p}SE(\beta_j),\hat{\beta_j}+t_{\alpha/2,n-p}SE(\beta_j)\} \\ & =\{\hat{\beta_j}-t_{\alpha/2,n-p}s\sqrt{(X'X)^{-1}_{jj}},\hat{\beta_j}+t_{\alpha/2,n-p}s\sqrt{(X'X)^{-1}_{jj}}\} \end{align*}$$` -- **Intepretation:** - We are `\(1-\alpha\)` % confident that the true parameter is within this CI - If we take repeated samples, `\(1-\alpha\)` % of such constructed confidence intervals would contain true `\(\beta\)` --- ### Example: For our age coefficient we had: - `\(\hat{\beta}_{Age}=0.206\)` - `\(SE(\hat{\beta})=0.067\)` - Our `\(n=5000\)` so we can use normal approximation -- So 95% CI for `\(\beta_{Age}\)` is: `$$\begin{align*} CI_{1-\alpha} & =\{\hat{\beta_j}-t_{\alpha/2,n-p}SE(\beta_j),\hat{\beta_j}+t_{\alpha/2,n-p}SE(\beta_j)\} \\ & =\{0.206-1.96*0.067,0.206+1.96*0.067\} \\ & =\{0.075,0.337\} \end{align*}$$` -- - Note that the CI does not contain 0 - What does it imply for hypothesis testing with `\(H_0: \beta_{age}=0\)`? --- ### CI for mean response .pull-left[ Suppose that we want an average prediction for individuals with these characteristics: `$$\mathbf{x_0}=\begin{bmatrix} 1 \\ x_{01} \\ x_{02} \\ \vdots \\ x_{0k} \\ \end{bmatrix}$$` Ex: What's average income ( `\(y\)` ), for people who whave 12 years of education `\(x_{01}=12\)` (2 other people are there) and are age 50 `\(x_{02}=50\)` How accurate is our prediction? `$$\hat{y}_0=\mathbf{x_0}'\hat{\beta}$$` ] .pull-right[ The prediction is unbiased: `\(E(\hat{y}_0)=\mathbf{x_0}'\beta\)` and it's variance is: `$$\begin{align*} var(\hat{y}_0)& =var(\mathbf{x_0}'\hat{\beta}) \\ & =\mathbf{x_0}'var(\hat{\beta})\mathbf{x_0} \\ &=\sigma^2\mathbf{x_0}'(\mathbf{X}'\mathbf{X})^{-1}\mathbf{x_0} \end{align*}$$` So it's distribution is: `\(\hat{y}_0 \sim N(\mathbf{x_0}'\beta, \sqrt{\sigma^2\mathbf{x_0}'(\mathbf{X}'\mathbf{X})^{-1}\mathbf{x_0}})\)` Hence: `\(CI_{1-\alpha}=\{\hat{y}_0\pm t_{n-2,\frac{\alpha}{2}}\sqrt{\sigma^2\mathbf{x_0}'(\mathbf{X}'\mathbf{X})^{-1}\mathbf{x_0}}\}\)` ] --- ### Exanmple What's the 95% CI for average wait time when there is 10 people at the Urgent Care `\(x_{occupancy}=10\)` for a person who is of age 52 `\(x_{age}=52\)` ? - What do we need to answer this question? - `\(\hat{\beta}=\{\hat{\beta_0}, \hat{\beta}_{occupancy}, \hat{\beta}_{age}\}=\{23.236, 3.7, 0.2\}\)` - `\(\sqrt{\mathbf{x_0}'(\mathbf{X}'\mathbf{X})^{-1}\mathbf{x_0}}=\sqrt{[1, 10, 52](\mathbf{X}'\mathbf{X})^{-1}[1, 10, 52]'}=0.021\)` - `\(\sigma=98.97\)` -- - Prediction: `\(\hat{y_0}=23.236*1+3.7*10+0.2*52=70.636\)` -- - Standard Deviation: `\(SE(\hat{y_0})=\sqrt{\sigma^2\mathbf{x_0}'(\mathbf{X}'\mathbf{X})^{-1}\mathbf{x_0}}=2.07837\)` -- `$$CI_{95}=\{70.636 \pm 1.96*2.07837\} \approx \{67, 75\}$$` --- ### Exanmple Or simply in R: ```r lm_model <- lm(Duration ~ Occupancy+EDAD, data = Sample_urg) new_data<- data.frame(Occupancy= c(10), EDAD=52) predict(lm_model, newdata = new_data, interval = "confidence", level = 0.95, se.fit=TRUE) ``` ``` ## $fit ## fit lwr upr ## 1 70.98457 66.92534 75.0438 ## ## $se.fit ## [1] 2.070572 ## ## $df ## [1] 4997 ## ## $residual.scale ## [1] 98.97419 ``` --- ### CI for new observation ** Reminder **: - When we look at average response, `\(u_i\)` doesn't play a role (because on average errors are 0) - When we look at a single observation, `\(u_i\)` matters, so it increases the variance of prediction error So variance is now the previous variance plus the variance of `\(u_i\)` `$$\begin{align*}var(y_0-\hat{y}_0)& =var(x_0\beta+u_i-x_0\hat{\beta}) \\ & =var(u_0)+var(x_0\hat{\beta}) \\ & =\sigma^2+\sigma^2\mathbf{x_0}'(\mathbf{X}'\mathbf{X})^{-1}\mathbf{x_0}\end{align*}$$` So the confidence interval for a single observation is slightly wider: `\(CI_{1-\alpha}=\{\hat{y}_0\pm t_{n-2,\frac{\alpha}{2}}\sqrt{\sigma^2(1+\mathbf{x_0}'(\mathbf{X}'\mathbf{X})^{-1}\mathbf{x_0})}\}\)` We are less certain about predicting outcome for a single person, compared to average outcome among namy people. --- ### Testing for multiple coefficients A cool thing about the regression is that we can test relationships between the coefficients: **For example**: - Is the impact of additional year of experience the same as impact of additional year of work experience in a regression: `$$income_i=\beta_0+\beta_1\text{education}_i+\beta_2\text{experience}_i+u_i$$` - That corresponds to null hypothesis `\(H_0: \beta_1=\beta_2\)` or `\(H_0: \beta_1-\beta_2=0\)` **Another Example**: - Suppose that employees can go through a sales training, get a promotion, and/or get better office (these are binary variables). We want to evaluate impact of these measures on their sales: `$$Sales_i=\beta_0+\beta_1\text{training}_i+\beta_2\text{promotion}_i+\beta_3\text{office}_i+u_i$$` - We wonder if giving an employee all three would increase sales by 100: `\(H_0: \beta_1+\beta_2+\beta_3=100\)` --- ### General Linear Hypothesis We can express all these null hypothesis in a general way as: `$$H_0: \mathbf{T}\beta=\mathbf{c}$$` where `\(\mathbf{T}\)` is a `\(r \times p\)` and `\(\mathbf{c}\)` is a vector of size `\(r\)` - r is number of restrictions the hypothesis impose on the parameters --- Consider this example: `$$income_i=\beta_0+\beta_1\text{education}_i+\beta_2\text{experience}_i+u_i$$` - Null hypothesis `\(H_0: \beta_1=\beta_2\)` or `\(H_0: \beta_1-\beta_2=0\)` -- - Here: $$T=\begin{bmatrix} 0 & 1 & -1 \end{bmatrix}$$ and `\(c=0\)` -- So that: `$$H_0: \mathbf{T}\beta=\begin{bmatrix} 0 & 1 & -1 \end{bmatrix}\begin{bmatrix} \beta_0 \\ \beta_1 \\ \beta_2 \end{bmatrix}=\beta_1-\beta_2=0$$` --- --- Consider this example: `$$Sales_i=\beta_0+\beta_1\text{training}_i+\beta_2\text{promotion}_i+\beta_3\text{office}_i+u_i$$` - We wonder if giving an employee all three would increase sales by 100: `\(H_0: \beta_1+\beta_2+\beta_3=100\)` -- - Here: $$T=\begin{bmatrix} 0 & 1 & 1 & 1 \end{bmatrix}$$ and `\(c=100\)` -- So that: `$$H_0: \mathbf{T}\beta=\begin{bmatrix} 0 & 1 & 1 & 1 \end{bmatrix}\begin{bmatrix} \beta_0 \\ \beta_1 \\ \beta_2 \\ \beta_3 \end{bmatrix}=\beta_1+\beta_2+\beta_3=\underbrace{100}_{c}$$` --- --- What if our hypothesis includes more than 1 restriction? `$$Sales_i=\beta_0+\beta_1\text{training}_i+\beta_2\text{promotion}_i+\beta_3\text{office}_i+u_i$$` - We wonder ig - Giving an employee an office has the same impact as giving her a promotion .blue[and] - Giving an employee an office has the same impact as giving training: -- - `\(H_0: \beta_1=\beta_2\text{ and }\beta_1=\beta_3\)` -- - Here we have two restrictions: `$$\mathbf{T}=\begin{bmatrix} 0 & 1 & -1 & 0 \\ 0 & 1 & 0 & -1 \end{bmatrix} \qquad c=\begin{bmatrix} 0 \\ 0 \end{bmatrix}$$` -- So that: `$$H_0: \mathbf{T}\beta=\begin{bmatrix} 0 & 1 & -1 & 0 \\ 0 & 1 & 0 & -1 \end{bmatrix}\begin{bmatrix} \beta_0 \\ \beta_1 \\ \beta_2 \\ \beta_3 \end{bmatrix}=\begin{bmatrix} \beta_1-\beta_2 \\ \beta_1-\beta_3 \end{bmatrix}=\underbrace{\begin{bmatrix} 0 \\ 0 \end{bmatrix}}_{c}$$` --- What about testing if any coefficient is significantly different from 0? `$$Sales_i=\beta_0+\beta_1\text{training}_i+\beta_2\text{promotion}_i+\beta_3\text{office}_i+u_i$$` -- - `\(H_0: \beta_1=0\text{ and } \beta_2=0\text{ and }\beta_3=0\)` -- - Here we have 3 restrictions: `$$\mathbf{T}=\begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{bmatrix} \qquad c=\begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$$` -- So that: `$$H_0: \mathbf{T}\beta=\begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{bmatrix}\begin{bmatrix} \beta_0 \\ \beta_1 \\ \beta_2 \\ \beta_3 \end{bmatrix}=\begin{bmatrix} \beta_1 \\ \beta_2 \\ \beta_3 \end{bmatrix}=\underbrace{\begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}}_{c}$$` --- ### General Linear Hypothesis - This framework allows us to do a bunch of cool tests! -- - **Intuition**: we compare errors in restricted model to errors in unrestricted model - Restricted model is where we force the null to be true -- - Consider this example: `$$income_i=\beta_0+\beta_1\text{education}_i+\beta_2\text{experience}_i+u_i$$` - Null hypothesis `\(H_0: \beta_1=\beta_2\)` - .red[Restricted Model]: `\(income_i=\beta_0+\beta_1\text{education}_i+\beta_1\text{experience}_i+u_i\)` - .green[Unrestricted Model]: `\(income_i=\beta_0+\beta_1\text{education}_i+\beta_2\text{experience}_i+\epsilon_i\)` -- - Suppose we estimate the two models: - If the **null hypothesis are true**, then the explained variation in restricted and unrestricted model should be similar - If the **null is not true**, explained variation in unrestricted model should be larger: because restricted model is wrong! --- We know how to get sum of squares and residuals from the unrestricted model But how do we do it for the restricted model?! - we will estimate model assuming that restrictions are true. - ex: estimate when h0 is that they are 0 - ex estimate this assuming they are eqaul You don't need to know full details, but T helps to do that --- So here is how it works ### General Linear Hypothesis significance of regression - all of them are 0 This corresponds to ANOVA Write down ANOVA TABLE --- Example in R --- ### General Linear Hypothesis Here is couple of examples: testing equality of coefficient equality of two coefficients --- Example in R --- ### Standarized Coefficients - Coefficients depend on the units of measurement of the `\(x\)` - Since `\(x\)` can have different units or magnitudes, we can't directly compare them -- **Example:** $$\text{ecobici trips}_i=\beta_0+\beta_1\text{temperature}_i+\beta_2\text{polution}_i+u_i $$ -- - It doesn't make sense to compare `\(\beta_1\)` to `\(\beta_2\)` to see what has bigger effect - These variables have very differentt magnitues - Increasing temperature by one unit (1 degree celcius) is different than increasing polution by one unit (1 μg/m3) -- - To make them directly comparable, we want to make them unitless (standarized) - Does increasing temperature by .blue[one standard deviation] has the same effect as inreasing polution by .blue[one standard deviation]? --- ### Standarized coeffcients Basically, we standardize all the variables and run the regression: `$$\frac{y_i-\bar{y}}{s_y}=\gamma_1\frac{x_{i1}-\bar{x}_{1}}{s_{{x}_1}}+\gamma_2\frac{x_{i2}-\bar{x}_2}{s_{{x}_2}}+...+\gamma_k\frac{x_{ik}-\bar{x}_k}{s_{{x}_k}}+u_i$$` So then `\(\gamma_k\)` measures the impact of one standard deviation increase of `\(x_k\)` on standard deviation in y -- But there is a short cut to calculate these standard coefficients `$$\gamma_k=\beta_k\frac{s_{{x}_k}}{s_y}$$` --- ### Example Urgent Care duration example: - `\(s_y=111.82\)` - `\(s_{Age}=20.82\)` - `\(s_{Occupancy}=13.921\)` -- We calculated that `\(\hat{\beta}_{Age}=0.206\)` and `\(\hat{\beta}_{Occupancy}=3.703\)` -- **Standardized coefficients** `$$\begin{align*} & \hat{\gamma}_{Age} =\hat{\beta}_{Age}\frac{s_{Age}}{s_{y}}=0.206\frac{20.82}{111.82}=0.0383 \\ &\hat{\gamma}_{Occupancy} =\hat{\beta}_{Occupancy}\frac{s_{Occupancy}}{s_y}=3.703\frac{13.921}{111.82}=0.461 \end{align*}$$` -- - Changing age by one standard deviation increases duration by 3.8% of a standard deviation - Changing occupancy one standard deviation increases duration by 46% of a standard deviation ---